Problem: The vertices of an equilateral triangle lie on the hyperbola $xy=1$, and a vertex of this hyperbola is the centroid of the triangle. What is the square of the area of the triangle?
Solution: Without loss of generality, suppose that the centroid of the triangle is at the vertex $(-1,-1)$.  In an equilateral triangle, the centroid and the circumcenter coincide, so the three vertices of the triangle are among the intersection points of the hyperbola $xy = 1$ and a circle centered at $(-1,-1)$.

Suppose the hyperbola and circle intersect at four distinct points, shown below on the left, at $A$, $B$, $C$, and $D$.  Either $A$ or $B$ are two of the vertices, or $C$ and $D$ are two of the vertices.  If $A$ and $B$ are two of the vertices, then the triangle will have the line $y = x$ as an axis of symmetry, which means that the third vertex must also lie on the line $y = x$.  However, neither of the other two points satisfy this condition.  The argument is the same if $C$ and $D$ are two of the vertices.

[asy]
unitsize(0.8 cm);

real f(real x) {
  return(1/x);
}

pair A, B, C, D, trans = (9,0);

A = intersectionpoints(Circle((-1,-1),3),graph(f,1/3,3))[0];
B = intersectionpoints(Circle((-1,-1),3),graph(f,1/3,3))[1];
C = intersectionpoints(Circle((-1,-1),3),graph(f,-5,-1/5))[0];
D = intersectionpoints(Circle((-1,-1),3),graph(f,-5,-1/5))[1];

draw((-5,0)--(3,0));
draw((0,-5)--(0,3));
draw(graph(f,1/3,3),red);
draw(graph(f,-1/5,-5),red);
draw(Circle((-1,-1),3));

dot("$A$", A, NE);
dot("$B$", B, NE);
dot("$C$", C, SW);
dot("$D$", D, SW);
dot("$(-1,-1)$", (-1,-1), SW);

draw(shift(trans)*((-5,0)--(3,0)));
draw(shift(trans)*((0,-5)--(0,3)));
draw(shift(trans)*graph(f,1/3,3),red);
draw(shift(trans)*graph(f,-1/5,-5),red);
draw(Circle((-1,-1) + trans,2*sqrt(2)));

dot("$(-1,-1)$", (-1,-1) + trans, SW);
dot("$(1,1)$", (1,1) + trans, NE);
[/asy]

Therefore, the hyperbola must intersect the circle at exactly three points.  In turn, the only way this can happen is if the circle passes through the point $(1,1)$.  The circumradius of the triangle is then the distance between $(-1,-1)$ and $(1,1)$, which is $2 \sqrt{2}$.  It follows that the side length of the triangle is $2 \sqrt{2} \cdot \sqrt{3} = 2 \sqrt{6}$, so the area of the triangle is $\frac{\sqrt{3}}{4} \cdot (2 \sqrt{6})^2 = 6 \sqrt{3},$ and the square of the area is $(6 \sqrt{3})^2 = \boxed{108}.$